Around the World in 80 Questions (Day 9): If n is a positive integer

Question

If n is a positive integer, what is the value of (x - 3)^n + (3 - x)^n ?

(1) n divided by 6 gives the remainder of 1.
(2) 21 divided by n gives the same remainder as 20 divided by n.

nivivacious · Answer

(1) n divided by 6 gives the remainder of 1. 
Therefore n is in the form 6k+1, which is an odd number. 
So value of (x - 3)^n + (3 - x)^n is 0.
Hence sufficient

(2) 21 divided by n gives the same remainder as 20 divided by n. 
This means n must be a factor of 1, because the difference between 21 and 20 is 1. Therefore, n can only be 1.
if n is 1, it is odd and the value of (x - 3)^n + (3 - x)^n is 0.
Hence sufficient

Hence D

vedha0 · Answer

answer is D.

STATEMENT 1 : n divided by 6 gives the remainder of 1.
so n is odd. (3-x)^n = ((-1)*(x-3))^n = (-1)^n * (x-3)^n = -((x-3)^n) ------ since n is odd
therefore the required value is 0. this statment is  sufficient 
 
STATEMENT 2 : 21 divided by n gives the same remainder as 20 divided by n.
21 and 20 are co prime. and n is positive. thus n is 1. placing 1 in the equation, we can determine the value to be 0. this statment is  sufficient

EACH ALONE IS SUFFICIENT. SO ANSWER IS D

missionmba2025 · Answer

(x - 3)^n + (3 - x)^n

(x - 3)^n + (-1)^n(x - 3)^n

(1) n divided by 6 gives the remainder of 1.

n is odd, hence (1)^n = -1

(x - 3)^n - (x - 3)^n = 0

The value is zero. Sufficient.

(2) 21 divided by n gives the same remainder as 20 divided by n.

20 and 21 are consecutive numbers, hence co-primes. Only common factor = 1

n is odd, hence (1)^n = -1

(x - 3)^n - (x - 3)^n = 0

Sufficient.

IMO D

Around the World in 80 Questions (Day 9): If n is a positive integer

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